Theorem iunin1 3872

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3861 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1	|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Distinct variable group:   x, B

Allowed substitution hints:   A( x)   C( x)

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 3871	. 2 |- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
2		incom 3263	. . . 4 |- ( C i^i B ) = ( B i^i C )
3	2	a1i 9	. . 3 |- ( x e. A -> ( C i^i B ) = ( B i^i C ) )
4	3	iuneq2i 3826	. 2 |- U_ x e. A ( C i^i B ) = U_ x e. A ( B i^i C )
5		incom 3263	. 2 |- ( U_ x e. A C i^i B ) = ( B i^i U_ x e. A C )
6	1, 4, 5	3eqtr4i 2168	1 |- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )

Syntax hints:   = wceq 1331   e. wcel 1480   i^i cin 3065   U_ciun 3808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-iun 3810

This theorem is referenced by:  2iunin  3874  tgrest  12327